Basel, Johannes Herwegen, 1533. Folio. (323x220 cm). Cont. full blind-tooled calf with a broad border of ornamental rolls with corner-pieces, inside which an oblique blind-tooled parallelogram and a rectangular tooled decoration, also with corner-pieces. Professionally rebacked in old style, w. seven raised bands blindstamped ornamentations to all compartments. Corners professionally and neatly restored. (12), 268; 115, (1) pp. incl. last page with large woodcut printer's device. Numerous woodcut diagrams in the text. The last page of Grynaeus' foreword with a half-page note on Euclid, Proclus and Grynaeus in 18th century hand. One contemporary marginal note. First 3 leaves with faint finger-soiling to lower right corner. The text framed throughout by a decorative but faint ink-border. Verso of title-page with 2 small stamps. Title with woodcut printer's device. The first text-page framed with a broad woodcut border, many smaller and larger woodcut initials throughout. Internally a very fine and clean copy w. wide margins.

The monumental editio princeps of the "Elements" of Euclid, "the greatest mathematical textbook of all times", being the first printing of the original Greek text, including the first printing of Proclus' seminal commentary to the first book (the so-called "Herwagiana"). The present editio princeps constitutes one of the most important publications in the history of scientific (and philosophical) thought, and it profoundly influenced Renaissance, and in turn all modern, thought. The first printing of the original Greek text of the "Elements", which is edited by the famous Basel-professor of Greek Simon Gryneaus the elder, served as the basis for all later texts and translations of the "Elements" until the nineteenth century. Proclus's seminal commentary to the first book, which had never been printed before, is considered the earliest contribution to the philosophy of mathematics and "one of the most valuable documents in ancient philosophy" (Morrow, p. XXXII). It profoundly influenced Renaissance and modern readings of Euclid's Elements and is responsible for the role that this magnum opus came to play during the Renaissance.

It is not until Proclus (ca. 410-485), the great Neoplatonist, applies Plato's manner of thinking to Greek geometry that it achieves completion as a real system. His view of mathematics as part of a larger system of thought was perfectly in tune with the currents of Renaissance thought, and with the commentary of Proclus, the Renaissance student of Euclid was carried beyond the ostensible boundaries of mathematics into the paths of cosmological and metaphysical speculation, paving the way for these fields in modern thought.

But Proclus' commentary is not only of seminal importance to the antique and Renaissance interpretation of the work, it also provides us with invaluable information regarding geometers and the history of geometry prior to Euclid. "Its numerous references to the views of Euclid's predecessors, many of them otherwise unknown to us, render it an invaluable source for the history of science." (DSB, pp. 160-61). "These numerous and sometimes very extended references to opinions and accomplishments of his predecessors, taken together with the material rescued from Eudemus's early history of geometry, make Proclus' "Commentary" a priceless source of information regarding the geometry of the previous nine or ten centuries." (Morrow. p. XXVIII). -"Yet the value of the matter it contains regarding the foundations of mathematics and geometry in particular is even greater, though less widely recognized." (Morrow, p. XXXII).

Proclus here explains the meaning of "Element" in geometry, he states the theoretical and pedagogical purposes of an elementary treatise, and offers a striking evaluation of the excellence of Euclid's own work. Futhermore, he famously defends pure mathematics, and geometry in particular, against its critics, and includes an important interpretation of the attitude of Plato, who was often used by these critics, against mathematics. Proclus furthermore raises questions that are absolutely fundamental to the understanding of both Plato and the science of Euclid, namely what the nature of the objects of mathematic enquiry is, and what the validity of the procedures used to handling them are. Posing these absolutely fundamental problems for the first time makes Proclus the first real philosopher of mathematics. "Proclus' treatise is the only systematic treatise that has come down to us from antiquity dealing with these questions". (Marrow, p. XXXIII).

Proclus' commentary, which takes up the second part of the book, pp. 1-115, is also known as the "Herwagiana", named after the printer. Apart from the above-mentioned elements of the commentary, it also constitutes the first criticism of Euclid to question the "Parallel-axiom", - hereby paving the road to "NON-EUCLIDEAN GEOMETRY". Proclus was the first commentator to be very explicit about his objection to the Parallel axiom, as he refused to count it among the postulates. To justify his opinion he remarks that the converse (the sum of two angles is less than that of two right angles), is one of the theorems proved by Euclid (Book I. Prop. 17), and he thinks it impossible that a theorem, the converse of which can be proved, is not itself capable of proof. He says: "This (postulate) ought even to be struck out of the postulates altogether; for it is a theorem involving many difficulties, which Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it a number of definitions as well as theorems, and the converse of it is actually proved by Euclid himself as a theorem." - Proclus' proof, taking up another axiom, was essentially correct, but he substituted one questionable axiom for another. (Se Bonola: Non-Euclidean Geometry).

It goes without saying that Euclid's treatise itself, the "Elements" also directly influenced all scientific thought ever since its appearance. The exemplary role of geometry after Euclid enjoyed uncontested supremacy for centuries, until the discovery of non-Euclidean geometry introduced entirely new questions for mathematical thought and forced it to a new interpretation of its own logical structure.

"There are few books that have played a larger part in the thought and education of the Western world than Euclid's "Elements". For more than twenty centuries it has been used as an introduction to geometry, and only within the last hundred years has it begun to be supplemented, or supplanted, by more modern textbooks. "This wonderful book", writes Sir Thomas Heath, "with all its imperfections, which indeed are slight enough when account is taken of the date at which it appeared, is and will doubtless remain the greatest mathematical textbook of all times. Scarcely any other book except the Bible can have been circulated more widely the world over, or been more edited and studied"." (Morrow, pp. XXI-XXII).

"The most famous source of Greek geometry is the monumental work of Euclid of Alexandria, called the "Elements" (around 300 B.C.). No other book of science had a comparable influence on the intellectual development of mankind. It was a treatise of geometry in thirteen books which included all the fundamental results of scientific geometry up to his time. Euclid did not claim for himself any particular discovery, he was merely a compiler. Yet, in view of the systematic arrangement of the subject matter and the exact logical procedure followed, we cannot doubt that he himself provided a large body of specific formulations and specific auxiliary theorems in his deductions. It is no longer possible to pass judgment on the authorship of much of this material; his book was meant as a textbook of geometry which paid attention to the material, while questions of priority did not enter the discussion" (Cornelius Lanczos in "Space through the Ages").